Talk:Diffeomorphism

WikiProject Mathematics (Rated C-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 C Class
 High Importance
Field: Geometry

HELP : can't get the links to Manifold#Differentiable_manifolds to work. When I click (try atlases for instance) I get to the top of the Manifold page. What's wrong ? Pascalromon 14:19, 27 Nov 2003 (UTC)

Unresolved question concerning unexplained mathematical symbols

I think I followed the definition at the beginning through and including this statement:

Two manifolds M and N are diffeomorphic (symbol being usually $\simeq$) if there is a diffeomorphism $f$ from M to N. For example

.... So far, so good, but wait ... The very next line has this expression:

$\mathbb{R}/\mathbb{Z} \simeq S^1.$

Let me take some guesses at what this means in layperson's terms:

• "The set of real numbers without the integers is diffeomorphic to $S^1$ (whatever $S^1$ means)" ... or is it something like:
• "The quotient set formed by the real numbers with the integers (see also word salad (mental health)) is diffeomorphic to $S^1$ (whatever $S^1$ means)"

If in about a week or so I see no disambiguation either here or in the article, then I plan to be bold and update the article with one of the above two definitions.

Please be gentle on me, my WikiStress is higher than usual. Vonkje 16:12, 1 Jun 2005 (UTC)

It's the quotient group. You can think of it this way: take a line segment and bend it round into a circle, identifying the end points. Of course that gives you one special point, and there should be no special points at all. Still, it is the correct topological picture. Charles Matthews 16:40, 1 Jun 2005 (UTC)

set minus is usually written with a backslash (or at least looks like one), not a forward slash. So the reals minus the integers would be $\mathbb{R}\setminus \mathbb{Z}$. As Charles points out, what we have here is$\mathbb{R}/\mathbb{Z}$, the quotient group of the additive group of real numbers modulo the normal subgroup of integers. This quotient group is set of all cosets of $\mathbb{Z}$. Cosets will look like {..., -2.5, -1.5, -0.5, 0.5, 1.5, 2.5,...}, which is the coset $0.5+\mathbb{Z}$. Notice that the cosets $0+\mathbb{Z}$ and $1+\mathbb{Z}$ are actually the same coset, so that this quotient space eventually gets back to where it started, just like a circle. Speaking of circles, $S^n$ or sometimes $\mathbb{S}^n$ is the n-dimensional sphere. A 1-dimensional sphere is also called a circle. So the observation that the quotient group above comes back to where it starts is the starting point of the observation that that quotient group is like a circle. If you see your circle as the set of complex numbers of modulus 1, then your diffeomorphism can be given by the formula $x\mapsto e^{2\pi ix}$. Then you just have to check that this is a homeomorphism, is smooth, and has smooth inverse. It will also turn out to be an isomorphism of groups, so that it is actually an isomorphism of Lie groups. Hope that helps. -Lethe | Talk 18:36, Jun 1, 2005 (UTC)
Thank you so much ... your added explanation in the article, and your patient explanation on this page helped. ... So where the cosets $0+\mathbb{Z}$ and $1+\mathbb{Z}$ are the same would refer to that "special point" that Charles was mentioning where the two ends of the line segment forming the circle meet? Vonkje 15:50, 6 Jun 2005 (UTC)
Charles's description has a special point because he started with a line segment. A line segment has two special points: its endpoints. When you join the two endpoints with glue, they become a single point, and a real circle doesn't have any special points, from a topological standpoint. But the true construction R/Z doesn't suffer this defect. Like I mentioned, 0+Z and 1+Z are the same cosets, and so the same points of R/Z, however, they are not special. It's also true that 0.5+Z and 1.5+Z are the same points, and 0.9+Z and 1.9+Z are the same points, etc. Since I started with an infinite line R, instead of a line segment, there are no special points anywhere, at least not from a topological point of view. When you recall that R and Z are groups, both have a special point, 0, the identity. R/Z then has a special point as well, but this is OK for groups. You could get rid of this special point by viewing R as an affine space with a Z action, and R/Z the set of Z-orbits. -lethe talk 21:23, 29 November 2005 (UTC)

Newcomer to the subject here, but isn't the diffeomorphism in the example above rather given by $x \mapsto e^{2\pi i x}$? Thanks, 155.198.196.187 11:03, 31 May 2006 (UTC)

Yes. -lethe talk + 12:25, 31 May 2006 (UTC)

Any chance of a lay example?

I'm linking into here from guage theory, trying to wrap my head around some of these topics. However, IMHO, all of them are written by math nerds for other math nerds. Can't anyone provide a single example of these concepts for us non-math-nerds? The second sentance of ANY article should not start "the mathematical definition is as follows"!

Maury 22:26, 28 November 2005 (UTC)

The definition is very simple. Just read carefully, it says a function which is invertible, and both the function and the inverse are smooth. I assume you know what the words "invertible function" and "smooth" mean. Otherwise, this article will not be helpful to you. Oleg Alexandrov (talk) 22:33, 28 November 2005 (UTC)
And there is an example there, and a good discussion at ==Local description==. Oleg Alexandrov (talk) 22:34, 28 November 2005 (UTC)

it says a function which is invertible, and both the function and the inverse are smooth

But it's buried. If this is the definition, it should be the second sentance of the article. The definition should be blocked off into its own para. Like this... Maury 13:06, 29 November 2005 (UTC)

I don't think it was buried. It was the third sentence in the article (if you count the wording "The mathematical definition is as follows." as a sentence). But I do agree that it looks better now. Oleg Alexandrov (talk) 20:18, 29 November 2005 (UTC)

R/Z : Divide any real number by any integer; throw away the integer part of the answer; what remains is a number, x, between -1 and +1: -1 < x < +1; this set of numbers is R/Z. Now use the formula, z(x) = exp( i*pi*x ); "exp" = "e to the"; "*" = multiply ); for any value of x in our set, R/Z, z(x) is a point on the unit circle in the Complex Plane. This is the circle centered on 0 (the origin), and having radius, 1. For example, the x-value 0 maps onto the point z=1; the x-value 0.5 maps onto the point z=i, the x-value 1 (we'll allow it to sneak in) maps onto the point z=-1, and the x-value -0.5 maps onto the point z=-i. So using this sequence of x-values we have traversed the unit circle counterclockwise, E, N, W, S. If we allowed the x-value, x=-1, it would also map to z=-1, since exp( i*pi*1) = exp( -i*pi*1) = -1 (Euler's identity, q.v.). This unit circle is called S1, because a circle is a 1-dimensional "sphere" (embedded in 2-dimensional space), a sphere (S2) is a 2-dimensional object (embedded in 3-dimensional space), etc. The integer part of the cosets (the "Z" part), when inserted into the formula, sends you to z=+1, z=-1, (E and W), again and again, contributing nothing new. So we have mapped a line segment (-1 to +1) on the Real axis, onto the unit circle, of which it is a diameter. If this segment is traversed from -1 to +1, then the unit circle will be traced from -1, counterclockwise, all the way around to -1 again.

I've added three nice examples. The example about the reals modulo the integers was too algebraic, and not very informative. Since all manifolds are locally diffeomorphic to real space we can do all practical calculations in coordinate charts. Thus I have given three examples of maps from two-space to two-space. For any calculation I have ever had to make (and I work in applications of singularity theory to differential geometry) I have always used the Jacobian method in my three examples. It's all well and good saying that a map and it's inverse must be differentiable, but how does one compute an explicit inverse for a map? With the Jacobian method it's automatic: it tells you that the map is differentiable, and that there exists a differentiable inverse. Dharma6662000 (talk) 19:24, 16 August 2008 (UTC)

Manifolds or just normed vector spaces?

The current definition of a diffeomorphism is given in terms of a map between manifolds, but a diffeomorphism can be defined over much more general spaces (namely, normed vector spaces - see e.g. Marsden, Ratiu, & Abraham, "Manifolds, Tensor Analysis, and Applications"). Why is the more specific case used here? Trevorgoodchild (talk) 05:23, 29 January 2008 (UTC)

I think the most general case is even more general than that, you can have diffeomorphisms over infinite-dimensional manifolds (which look locally like a normed vector space). But I guess going that general could make the article hard to understand. The most used cases are in manifolds over R^n anyway. If you want, you can add somewhere at the bottom a section on generalizations, but I'd prefer that at least the first several sections in the article be about the "usual" R^n manifolds case. Oleg Alexandrov (talk) 15:44, 29 January 2008 (UTC)

Relationship with homoeomorphisms

The "model example" section says that the diffeomorphism $f$ "happens to be" a homoeomorphism. From the definition, it seems to me that all diffeomorphisms are homoeomorphisms, since all differentiable functions are continuous. If that is the case, "happens to be" seems misleading. LachlanA (talk) 22:45, 6 May 2008 (UTC)

Yeah you're totally right: every diffeomorphism is a homeomorphism, but the converse is false. I'm sure you knew, but just to make sure: a function and its inverse must be differentiable (resp. continuous) for it to be a diffeomorphism (resp. homeomorphism). I've changed the article, and made your point clear. I replaced the sentence in question with: "f being a diffeomorphism is a stronger condition than f being a homeomorphism. For a diffeomorphism we need f and its inverse to be differentiable. For a homeomorphism we only require that f and its inverse be continuous. Dharma6662000 (talk) 19:28, 16 August 2008 (UTC)

Diffeomorphic manifolds=a smooth bijection with smooth inverse between them

I am quite sure that two manifolds are diffeomorphic iff there is a smooth bijective function between them that also has a smooth inverse (not as in the article where only a bijective differentiable function with differentiable inverse is required). For example, think about it this way: two mathematical objects are equivalent (and considered the same) iff there is an isomorphism between them. According to the definition given in the article, two differentiable manifolds may be equivalent without really having the same differentiable structure.

In fact, C0 functions are just continuous, so according to the article, two manifolds are diffeomorphic if they are homeomorphic. I am therefore changing this. If you think otherwise, please give your reasons.

Topology Expert (talk) 14:03, 6 October 2008 (UTC)

For connected manifolds, the diffeomorphism group acts transitively on the manifold???

We need some experts on diffeomorphism groups. Currently, the content has no meaning and is incorrect. For instance, the article claims the title of this section. I wanted to believe it but it seems somewhat dodgy. Here is a counterexample:

The line with two origins or the bug-eyed line; see [1]. This space is trivially connected. However, there is no homeomorphism carrying (0,b) -> [(5,b), (5,a)] (for instance). Suppose such a homeomorphism exists; let y denote the image of (0,a) under this homeomorphism and x denote the image of (0,b) under this homeomorphism. Clearly, x is not y and there are disjoint neighbourhoods about x and y respectively (the space is Hausdorff apart from (0,a) and (0,b)). Continuity of the homemorphism means that there are disjoint neighbourhoods about (0,a) and (0,b); a contradiction. Since every diffeomorphism is a homeomorphism, this is a counterexample to the claim in the article.

I am not an expert on the subject and I don't know of many of the important results to do with diffeomorphism groups. If someone knows, could he/she please include them in the article (provided there is a reference)?

Another issue is the factual accuracy of the article. The article claimed (or said in effect that):

• A curve with a cusp, in particular the curve, x^2 - y^3 = 0, is not difeomorphic to a straight line. Wrong, since the projection map yields a diffeomorphism (if you consider the natural differential structures on the two manifolds)
• A fibre bundle is a space (which is wrong); fibre bundles are ordered quadruples (if you don't consider structure groups acting on them)

I fixed both these things and I will remove the statement given as the title of this section. If there are any objections, please post your reasons here.

Topology Expert (talk) 03:16, 9 October 2008 (UTC)

Hi there. Manifolds in this article (as standard on Wikipedia unless otherwise noted) are considered to be 2nd countable Hausdorff. Although the diffeomorphism groups of non-Hausdorff manifolds are occasionally considered in research literature, more often than not they're considered beside-the-point. Generally speaking the people who work on diffeomorphism groups are living in the Hausdorff world, and if they do consider non-Hausdorff manifolds they make a point of it (rather than the other way around that you're considering).

Hi Rybu,

There was quite a big discussion on what manifolds should be according to Wikipedia here. Also see manifold (the lede). In Wikipedia (in general at least), unless it is stated that manifolds must be Hausdorff in a given article, one can assume otherwise (I actually would prefer to assume that a manifold is second countable and Hausdorff but Wikipedia has separate articles for seperate assumptions of what a manifold is). If you assume the Hausdorff condition, just add it to the lede and I can add back that example.

Topology Expert (talk) 02:23, 10 October 2008 (UTC)

FYI, the curve $x^2 - y^3 = 0$ is not diffeomorphic to the real line, because this curve is not a submanifold. The proof follows from the implicit function theorem -- if it was a submanifold, you'd be able to write it as the graph of a smooth function $y=f(x)$ or $x=f(y)$ near the origin, but in the first case $f$ is not smooth, and in the 2nd case $f$ would have to be multi-valued.. contradiction. Did you erase that example? If so it'd be best to change it back as it's an important example. You've been making so many modifications it's hard for me to keep track. I'll see if I can find it in the revision history. Rybu (talk) 19:46, 9 October 2008 (UTC)

I agree, the curve is trivially not an immersed submanifold (at (0,0), the differential is not defined). Nor can it be an embedded submanifold. But the article wrote that this curve is not diffeomorphic to the straightline and according to your reason, this does not make any sense (since you can't speak of whether two curves are diffeomorphic unless they are both smooth manifolds).

I might be confused, but I am quite sure that it is still possible to give the curve a differentiable structure since the projection map yields a local homeomorphism with the curve and the reals (and the transition maps are smooth since there is only one (the identity map of R)). So with this (natural) differentiable structure, the curve is diffeomorphic to the straight line.

Any continuum has a smooth structure making it diffeomorphic to $\mathbb R^n$ but it's not a natural smooth structure. In this sense the smooth structure you're putting on the above curve would be considered unnatural. The natural smooth structure on a subset of a manifold is the one coming from the ambient manifold. This is a common notion for sets as well as submanifolds. ie: A function $f: X \to Y$ is smooth (for $X,Y$ subsets of manifolds) if for every point in $X$, $f$ admits a local smooth extension to an open neighbourhood of the point in the manifold. I elaborated on this in diffeomorphisms of subsets of manifolds under the main definition in the article. If you'd like a reference, see for example the Guillemin and Pollack textbook Differential topology. Rybu (talk) 04:58, 10 October 2008 (UTC)

Topology Expert (talk) 02:23, 10 October 2008 (UTC)

By the way, I noticed that you added that example back. But with manifolds, the definition of a map d being smooth is when f o d o f^(-1) is smooth (where f and f^(-1) are charts over a particular open set). The curve in question is a smooth 1-manifold so shouldn't the projection map's inverse be smooth? You seem to assume that unless the curve is a submanifold of R2 it can't be a smooth manifold (which is not true from what I know (I could be wrong though)). Also, the article contradicts itself since it says that for dimensions 1, 2 and 3, homeomorphic smooth manifolds must be diffeomorphic ([2], this is a contradiction if you assume that the curve is a smooth manifold which it is with the differentiable structure that I mentioned). But first could you please tell me why that curve is not a smooth manifold with that differentiable structure? Topology Expert (talk) 02:35, 10 October 2008 (UTC)

There's a bit of a problem with conventions going on here. As I mentioned above, smoothness of a map is something defined for arbitrary subsets of manifolds. I mention the curve is not a submanifold, I do not say that it's impossible to make it into a manifold via a creative choice of a homeomorphism with $\mathbb R$. Being a submanifold and being possible to turn a set into a manifold, are two completely different things. The 1st is relevant, the 2nd is not. The submanifold needs some cleaning up, but according to Wikipedia I'm using the word submanifold in the intrinsic embedded submanifold sense. Rybu (talk) 04:58, 10 October 2008 (UTC)

Taking a standard definition of manifolds, where 1-manifolds are either intervals or the circle (cf Milnor's "Topology from the Differentiable Viewpoint", Guillemin & Pollack, etc), the statement is that in dimension > 1 the group of diffeomorphisms of compact support acts n-transitively on the manifold. This is discussed in Augustin Banyaga's book for example. For the circle, ordering of points must be taken into account. A quick search of mathscinet gave the generalisations of Boothy, Hatakeyama and Michor & Vizman to symplectic and contact manifolds. Here it seems that, rather than taking an exotic definition of smooth manifold, compact or not, the assumptions of the original authors in the literature need to be unearthed. What is true is that the old article does not give page numbers in the sources, or any direct citations. This would help avoid situations like this.

Banyaga's book is available here. The transitivity lemma is stated and proved on Page 29. It gives the n-transitivity result for smooth manifolds of dimension >1. The assumptions on the manifolds are given in the first chapter of the book. An analogous result for symplectomorphisms appears on page 109. Mathsci (talk) 07:30, 10 October 2008 (UTC)

I have added a Notes section in the article (for some strange reason the Hirsh and Smale references had been left orphaned), together with a direct link to Banyaga for the point on transitivity. I added the comments on Choquet and Rado, having sourced them in the literature. These days I would actually add the exact references for both statements, time permitting. Mathsci (talk) 07:39, 10 October 2008 (UTC)

Dear Rybu,

I am not disagreeing with you but I usually assume differentiable manifolds to take any appropriate differentiable structure. I think that this is the point of view taken by most mathematicians. Of course, in the context of submanifolds, this may be different. However, prior to me deleting the example (the one with the curve having a cusp), the article had not mentioned anywhere that the reason that this curve is not diffeomorphic to a straight line is because one is not a smooth manifold. Furthermore, they did not state why the curve with the cusp is not a smooth manifold by referring to submanifolds. Now what you have written has the appropriate hypothesis so I am not going to remove it again.

I have got to go know (I have more to say but I might not be able to respond very soon; if I do I will let you know on your talk page).

Topology Expert (talk) 11:02, 10 October 2008 (UTC)

Accoding to Rybu, the curve with a cusp is not a submanifold of $R^2$? It is definitely an imbedded submanifold if not an immersed one. Furthermore, according to Rybu, the projection map from the curve with a the cusp to the x-axis (along wih the obvious atlas) is not natural? Why not (it would be 'natural' if we embedded the 'curve with a cusp' in itself (with the previous mentioned differential structure))? To determine whether two manifolds M and N are diffeomorphic, we should not have to know where they are imbedded in. I don't think that the reason given in the article is appropriate; the curve with the cusp is a submanifold of $R^2$ (although not an immersed one). So remove the example altogether (the current reason is bogus):

Saying that two spaces are not diffeomorphic because one of the spaces is not even a manifold is useless and unecessary. If one of them is not a manifold, there should not be any question as two whether they are diffeomorphic. Therefore, I will remove that example.

Topology Expert (talk) 08:58, 15 November 2008 (UTC)

This is an issue of standards. From the point of view of Differential Topology, your curve with a cusp has no natural manifold structure. You can make it a manifold in many ways, the way you've chosen (writing it as the graph of a non-smooth function and then mandating that it has the differentiable structure to make the function a diffeomorphism onto its image) allows one to make sense of a smooth structure, but it's in no way natural, because if you make a different choice of bijection with the reals, you can get a different smooth structure. To be precise -- in your terminology, there is no obvious atlas. What is the obvious atlas on a Cantor set, or any continuum? If you have no standards for what the words you use mean, it's not clear they belong in a mathematics article. I suppose undefined terms have a place, but here we're talking about something technical, not heuristic. Ah well. Somebody else can clean this up. Rybu (talk) 10:02, 15 November 2008 (UTC)

Maybe I 'clouded up' my main point with other discussion. My main point was:

As a subspace of itelf (with the projection map as a chart over the whole set), the 'curve with the cusp' is an immersed submanifold. Surely this is natural (and it is imbedded in itelf as it is in the natural manifold $R^2$)?

Topology Expert (talk) 12:45, 15 November 2008 (UTC)

Possible error in text

A differentiable bijection is not necessarily a diffeomorphism, e.g. $f(x)=x^3$ is not a diffeomorphism from to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.

Possibly this should be:

A differentiable bijection is not necessarily a diffeomorphism, e.g. $f(x)=x^3$ is not a diffeomorphism from its image to itself because its derivative equals 0 at x = 0 (and hence its inverse is not differentiable at x = 0). This is an example of a homeomorphism that is not a diffeomorphism. See the inverse function theorem.

--Alephtwo (talk) 17:52, 7 May 2009 (UTC)

The text is fine, you're misquoting it though. Apparently the blackboard bold R is not showing up on your end -- some kind of rendering problem in your browser, maybe? $\mathbb R$ (\mathbb R) is the TeX code. Rybu (talk) 20:03, 7 May 2009 (UTC)
Rybu, Thanks for catching that problem on my side, my browser did not render the blackboard bold R. The technical part of the statement that I question, is the statement: "derivative vanishes at 0". I believe that the derivative of $f(x)=x^3$ is defined at 0 ( f '(0) = 0 ). It is the derivative of the inverse function that cannot be defined at zero, because the zero value of the derivative of the original function f '(), is in the denominator of the of derivative of the the inverse function. From inverse function theorem :
Statement of the theorem
For functions of a single variable, the theorem states that, if $f$ is a continuously differentiable function and $f$ has nonzero derivative at a, then $f$ is invertible in a neighborhood of a, the inverse is continuously differentiable, ...
(aside, how do I make the derivative apostrophe display more clearly: $f'()$ ?
thanks
--Alephtwo (talk) 15:05, 8 May 2009 (UTC)
The derivative a diffeomorphism has to be invertible at every point. Vanishing in this elementary case precludes that possibility. As for the TeX question: $f^\prime$.I don't know if this helps. Mathsci (talk) 15:21, 8 May 2009 (UTC)
Alephtwo, there's no problem in the text. Perhaps it's a terminology issue. When someone says the derivative vanishes they mean the derivative exists but is zero. This implies (via the inverse function theorem) that f is not a diffeomorphism. So I'm happy with the text as-is. Rybu (talk) 16:39, 8 May 2009 (UTC)

Dubious

I flagged the statement

In both cases, the diffeomorphism group is locally homeomorphic to the space of Cr vector fields on the manifold (where r is the order of differentiability considered).

as dubious. To the best of my knowledge, there are diffeomorphisms in any neighborhood of the identity that are not in the image of the exponential map. Sławomir Biały (talk) 14:13, 4 December 2009 (UTC)

In any case, it seems unlikely that this can be true in both cases, as asserted in the article: the strong topology is not locally homeomorphic to the weak topology on a non-compact manifold. It seems like a way forward would be to write a short section that describes each topology a little bit more carefully, making sure also to provide references in order to verify the content and also to avoid possible misunderstandings. Sławomir Biały (talk) 16:44, 4 December 2009 (UTC)
I think I have resolved the problem. The article wasn't claiming that the local homeomorphism came from the exponential mapping of the group. Sławomir Biały (talk) 02:53, 5 December 2009 (UTC)

The following content was recently added to the text. I feel that it is out of place there because the first section does not clearly say what it is trying to do (namely that the Lie algebra of the diffeomorphism group is the space of vector fields, which can be mentioned in a better way elsewhere). The addition of a fermion is poorly explained, and does not show how one gets a "group" out of the resulting Lie superalgebra. The second section seems to be about diffeomorphism groups in a particular context (relativity theory maybe), and so again seems poorly placed and poorly explained. I'm leaving this here for discussion, since it's possible some of this content should be worked into the article in some other way. Sławomir Biały (talk) 11:52, 22 April 2012 (UTC)

Update: I have incorporated some of the content about the Lie algebra into an earlier paragraph of the Diffeomorphism group section. There should be a separate article on superdiffeomorphism group that includes noncommutativity. I think this has little place in an article about the diffeomorphism group of an (ordinary) manifold. Sławomir Biały (talk) 12:00, 22 April 2012 (UTC)
=== Representations ===

By making a small change to the coordinate x at each point in space:

$x^{\mu} \rightarrow x^{\mu} + \varepsilon h^{\mu}(x)$

a simple representation of the diffeomorphism group is given by the (infinitesimal) generators is found:

$L_{h} = h^{\mu}(x)\frac{\partial}{\partial x_\mu}$

which form a group since:

$\left[ L_{h} , L_{f} \right] = L_{h.\nabla f - f.\nabla h }$

There is also a vector' matrix representation:

$(L_{h})^{\mu}_{\nu} = \delta^{\mu}_{\nu} h^{\tau}(x)\frac{\partial}{\partial x_\tau} - \frac{\partial h^{\mu}(x)}{\partial x_\nu}$

and a spinor' matrix representation:

$(L_{h})^{\alpha \beta} = \delta^{\alpha \beta} h^{\tau}(x)\frac{\partial}{\partial x_\tau} - \frac{\partial h^{\mu}(x)}{\partial x_\nu} \sigma_{\mu \nu}^{\alpha \beta}$

and there are representations for tensors and mixed indices. The diffeomorphism group can be extended by supersymmetry by adding a new fermion parameter in which case the rule is:

$\left[ L_{h,\psi} , L_{f,\eta} \right] = L_{h.\nabla f - f.\nabla h + \psi \Gamma \eta , h.\nabla \eta - f.\nabla \psi}$
=== Subgroups ===

Subgroups of the diffeomorphism group include the conformal group and (in 4 dimensions) the Poincare group. For example the algbera for the generators of the Poincare group M and P can be found by substituting in:

$h^{\mu}(x) = M^{\mu}_{\nu} x^{\nu} + P^{\mu}$

Question on notation

$g_{|U \cap X} = f_{|U \cap X}$